A Generalized Linear Model for an Estimation of Drug Expenditures

1
A Generalized Linear Modelfor an Estimation
ofDrug Expenditures

• Supon Limwattananon, BSc (Pharm), MPHM, PhD
• Faculty of Pharmaceutical Sciences,
• Khon Kaen University, THAILAND

2
Objective
To demonstrate a generalized linear model (GLM),
a statistical approach appropriate for an
estimation of non-normally distributed drug
expenditures as explained by policy change and
other explanatory variables
Data source Longitudinal data on drug use in a
panel of 6,794 adults with chronic asthma in 17
hospitals for 3 years (2000 2002) Research
question What is the magnitude of effect of
2002-Universal Coverage (UC) Policy on
expenditures of drug use in patients with chronic
asthma ?
3
Drug Utilization Study Research Issues

• For a descriptive research
• To explain variation in patterns of drug use
• ? variation in drug expenditures
• Such variation is probably conditional on policy
  change
• or interventions given
• For statistical inference of the
  policy/intervention effect
• A need for multivariate models,
• controlled for other explanatory variables
  or covariates

4
Year as an Indicator for Policy Change
Use of expensive inhaled corticosteroids
Patient demographics
Annualized Expenditure for antiasthmatics
Health insurance schemes
Years of drug use
UC Policy
Hospital indicators
5
Statistical Model for an Estimation of
Expenditure for Antiasthmatics
Left hand side, dependent variable
(Yi) Expenditure for antiasthmatics per year in a
given patient Right hand side, explanatory
variables (Xi) ICSi Receiving inhaled
corticosteroids (ICS) in a year (1Yes,
0No) AGEi Age groups (AGE36 36-50 yr., AGE51
gt50 yr., vs. 18-35 yr.) SEXi Male vs.
Female SCHEMEi Health insurance schemes (CSMB,
UCLIC, UCROP, ROP vs. SS) YEARi Years of
antiasthmatic use (YR2001, YR2002 vs.
YR2000) HOSPi Indicator variables for 17 study
hospitals ?i Unexplained portion of the
expenditures in the specified model
6
Longitudinal Data for Drug Use Study Methodologica
l Issues
Longitudinal data on use of drugs For a
given patient, there are multiple Rx visits or
drug use in a year Patient-year as a unit of
analysis Aggregation ? Annualized expenditures
per patient

• Behavior of expenditure data
• Such annualized expenditures tend to vary a lot
  across patients
• Such expenditures vary a lot more in groups with
  high expenditures
• Variation in the expenditures is reflected by
  variance of ?i

7
Classical Linear Regression Ordinary Least
Squares (OLS) Method
To estimate beta coefficients (?), using a CLR
model (? magnitude of the effect on Y
of each X) Required assumption well-behaved
distribution of data 1. Normality (E?i
0) 2. Homoscedasticity (uniform variance of
?i) 3. Independence of ?i 4. Linearity ?i
Unexplained portion of the expenditures in the
specified model
8
Distribution of Drug Expenditures (N 6,794 in
Year 2002)
Mean 2,493 Baht Median 1,540 Baht Skewness
2.57 (P lt 0.001)
9
Distribution of Log-transformed Expenditures (N
6,794 in Year 2002)
10
Results Regression for Log-transformed Expenditure
. regress LnBaht ICS age male CSMB - ROP YR
HOSP
--------------------------------------------------
---------------------------- LnBaht
Coef. Std. Err. t Pgtt 95 Conf.
Interval ---------------------------------------
--------------------------------------
ICS 1.434188 .0196845 72.86 0.000
1.395605 1.472771 age36 .1495537
.0289291 5.17 0.000 .0928502
.2062571 age51 .3315848 .0281622
11.77 0.000 .2763845 .3867851
male .3524408 .0170663 20.65 0.000
.3189895 .3858921 CSMB -.0160937
.0318193 -0.51 0.613 -.0784622
.0462748 UCLIC -.093156 .0302468
-3.08 0.002 -.1524422 -.0338699
UCROP -.0935148 .0348599 -2.68 0.007
-.1618431 -.0251865 ROP -.3409096
.041128 -8.29 0.000 -.4215239
-.2602952 YR2001 .3201426 .0203597
15.72 0.000 .2802358 .3600493
YR2002 .091071 .0203806 4.47 0.000
.0511234 .1310186
11
Families of Data Distribution (Variance Var.Y
and Mean EY)
Family Variance function Binomial Var.Y
(EY) (1 - EY) Gaussian (normal) Var.Y
Constant Poisson Var.Y EY Gamma
Var.Y (EY)2
Expenditures in the high-cost groups tend to vary
a lot more than in the lower ones
12
Cook Book for Diagnosis Approach
Regress Y on explanatory variables
Save Yi hat (predicted Yi) Mean
Save ?i (studentized residuals)
Squared ?i Variance
ln(?i2) Log variance
ln(Yi hat) Log mean
Regress log variance on log mean
13
Results Variance Function of the Mean
. regress LnSqrRes LnYhat ------------------------
--------------------------------------------------
---- LnSqrRes Coef. Std. Err. t
Pgtt 95 Conf. Interval ---------------
--------------------------------------------------
------------ LnYhat 1.290466 .0225882
57.13 0.000 1.246192 1.334741
_cons -11.63317 .1727979 -67.32 0.000
-11.97187 -11.29447 ----------------------------
--------------------------------------------------
Ln(Var. Y) 1.3Ln(EY) Var. Y EY1.3
14
Variance Function of the Mean Other Drug Classes
Drug class Power of Mean ? Variance ACE
inhibitors 1.8 ACE inhibitors A2 receptor
antagonists 1.6 Calcium channel
blockers 1.5 Statins fibrates 1.3 NSAIDs
COX2 inhibitors 1.5 H2 antagonists proton
pump inhibitors 1.5 Antiretrovirals 1.7 Antiep
ileptics 1.6 Source Limwattananon S, et al.
Cost and Utilization Patterns of Drugs Prescribed
to Hospital-Visited Patients an Impact of
Universal Health Coverage Policy, 2003
15
Estimation Approach Generalized Linear Model
. glm Baht ICS age male CSMB - ROP YR HOSP,
family(gamma) link(log)
--------------------------------------------------
---------------------------- Baht
Coef. Std. Err. z Pgtz 95 Conf.
Interval ---------------------------------------
--------------------------------------
ICS 1.07386 .0163599 65.64 0.000
1.041796 1.105925 age36 .1386217
.0236636 5.86 0.000 .0922419
.1850015 age51 .2623115 .0232358
11.29 0.000 .2167701 .3078528
male .2951559 .0141285 20.89 0.000
.2674646 .3228471 CSMB .0410909
.0262891 1.56 0.118 -.0104348
.0926165 UCLIC -.0687261 .0248604
-2.76 0.006 -.1174515 -.0200006
UCROP -.0936884 .0287291 -3.26 0.001
-.1499963 -.0373804 ROP -.3036286
.034236 -8.87 0.000 -.3707299
-.2365274 YR2001 .1986592 .0167031
11.89 0.000 .1659218 .2313966
YR2002 .0566952 .0168355 3.37 0.001
.0236982 .0896921
16
Interpretation of Beta-coefficient (Semi-logarithm
ic Functional Form ln Y ? X ?)
For a continuous variable X ? Percentage
effect on Y per unit change in X For an
indicator variable X exp(?) 1 Percentage
effect on Y of a change in X from 0 to 1
status Ref Halvorsen R and Palmquist R. The
interpretation of dummy variables in
semilogarithmic equations. American Economic
Review 1980 70 474-475. Kennedy P.
Estimation with correctly interpreted dummy
variables in semilogarithmic equations.
American Economic Review 1981 71 802.
17
Explanatory variable difference in expenditure
for antiasthmatics Point estimate Lower
95 CI Upper 95 CI
18
Comparison between GLM and CLR
GLM CLR (gamma, log
link) (OLS on log expenditure)
19
Useful Readings
1. Blough DK, Ramsey SD. Using generalized
linear models to assess medical care costs.
Health Services and Outcomes Research
Methodology. 2000 1 185-202. 2. Blough DK,
Madden CW, Hornbrook MC. Modeling risk using
generalized linear models. Journal of Health
Economics 1999 18 153-171. 3. Manning WG. The
logged dependent variable, heteroscedasticity,
and the retransformation problem. Journal of
Health Economics 1998 17 283-295. 4. Manning
WG, Mullahy J. Estimating log models to
transform or not to transform? Journal of
Health Economics 2001 17 461-494.